If every principal left ideal of a ring is projective, the ring is called a ‘left p.p. ring’. It is known that a ring \(R\) is left semihereditary (i.e. every finitely generated left ideal is projective) if and only if every matrix ring over \(R\) is a left p.p. ring. Moreover, a ring \(R\) is von Neumann regular if and only if every upper triangular matrix ring over \(R\) is a left p.p. ring. These two results motivate the question addressed here: Is every structural matrix ring over a regular ring a left p.p. ring?
The authors show that in general this is not the case and then determine exactly for which structural matrix rings this will be true.